Question: You have found the following ages (in years) of 4 lions. Those lions were randomly selected from the 22 lions at your local zoo: $ 1,\enspace 6,\enspace 1,\enspace 1$ Based on your sample, what is the average age of the lions? What is the standard deviation? You may round your answers to the nearest tenth.
Solution: Because we only have data for a small sample of the 22 lions, we are only able to estimate the population mean and standard deviation by finding the sample mean $({\overline{x}})$ and sample standard deviation $({s})$ To find the sample mean , add up the values of all $4$ samples and divide by $4$ $ {\overline{x}} = \dfrac{\sum\limits_{i=1}^{{n}} x_i}{{n}} = \dfrac{\sum\limits_{i=1}^{{4}} x_i}{{4}} $ $ {\overline{x}} = \dfrac{1 + 6 + 1 + 1}{{4}} = {2.3\text{ years old}} $ Find the squared deviations from the mean for each sample. Since we don't know the population mean, estimate the mean by using the sample mean we just calculated {1.69} + {13.69} + {1.69} + {1.69}} {{4 - 1}} $ {s^2} = \dfrac{{18.76}}{{3}} = {6.25\text{ years}^2} $ As you might guess from the notation, the sample standard deviation $({s})$ is found by taking the square root of the sample variance $({s^2})$ ${s} = \sqrt{{s^2}}$ $ {s} = \sqrt{{6.25\text{ years}^2}} = {2.5\text{ years}} $ We can estimate that the average lion at the zoo is 2.3 years old. There is also a standard deviation of 2.5 years.